The moment problem:
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| Format: | Book |
| Language: | English |
| Published: |
Cham
Springer
[2017]
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| Series: | Graduate texts in mathematics
277 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke Literaturverzeichnis Seite 517-526 |
| Physical Description: | xii, 535 Seiten Illustrationen |
| ISBN: | 9783319645452 3319645455 9783319878171 |
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MARC
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Record in the Search Index
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|---|---|
| adam_text | Contents
1 Integral Representations of Linear Functionals ................... 13
LI Integral Representations of Functionals on Adapted Spaces. 14
1.1.1 Moment Functionals and Adapted Spaces.............. 14
1.1.2 Existence of Integral Representations................ 15
1.1.3 The Set of Representing Measures................... 19
1.2 Integral Representations of Functionals on
Finite-Dimensional Spaces.................................. 22
1.2.1 Atomic Measures.................................... 23
1.2.2 Strictly Positive Linear Functionals ................ 25
1.2.3 Sets of Atoms and Determinate Moment Functionals___ 27
1.2.4 Supporting Hyperplanes of the Cone of Moment
Functionals.......................................... 30
1.2.5 The Set of Atoms and the Core Variety ............. 34
1.2.6 Extremal Values of Integrals with Moment
Constraints.......................................... 36
1.3 Exercises................................................... 40
1.4 Notes....................................................... 41
2 Moment Problems on Abelian «»-Semigroups.......................... 43
2.1 «»-Algebras and «»-Semigroups............................... 43
2.2 Commutative «»-Algebras and Abelian «»-Semigroups........... 46
2.3 Examples.................................................... 49
2.3.1 Example 1: Ng, n* = n.............................. 49
2.3.2 Example 2; INq*, (n, m)* = (m, n).................... 51
2.3.3 Example 3: 7Ld, n* = —n.............................. 52
2.3.4 Example 4: Z,n* = n.................................. 53
2.4 Exercises................................................... 53
2.5 Notes..................................................... 54
vii
viii
Contents
Part I The One-Dimensional Moment Problem
3 One-Dimensional Moment Problems on Intervals: Existence........ 57
3.1 Positive Polynomials on Intervals........................... 57
3.2 The Moment Problem on Intervals............................... 63
3.3 The Symmetric Hamburger Moment Problem and Stieltjes
Moment Problem................................................ 67
3.4 Positive Polynomials on Intervals (Revisited)............... 69
3.5 Exercises..................................................... 75
3.6 Notes......................................................... 77
4 One-Dimensional Moment Problems: Determinacy.......................... 79
4.1 Measures Supported on Bounded Intervals....................... 79
4.2 Carleman ’s Condition......................................... 80
4.3 Krein’s Condition........................................... 85
4.4 Exercises..................................................... 90
4.5 Notes..................................................— 91
5 Orthogonal Polynomials and Jacobi Operators........................... 93
5.1 Definitions of Orthogonal Polynomials and Explicit Formulas... 94
5.2 Three Term Recurrence Relations............................... 98
5.3 The Moment Problem and Jacobi Operators...................... 102
5.4 Polynomials of the Second Kind............................... 104
5.5 The Wronskian and Some Useful Identities..................... 108
5.6 Zeros of Orthogonal Polynomials.............................. 112
5.7 Symmetric Moment Problems.................................... 115
5.8 Exercises.................................................... 117
5.9 Notes........................................................ 119
6 The Operator-Theoretic Approach to the Hamburger Moment
Problem............................................................. 121
6.1 Existence of Solutions of the Hamburger Moment Problem... 121
6.2 The Adjoint of the Jacobi Operator........................... 123
6.3 Determinacy of the Hamburger Moment Problem.................. 125
6.4 Determinacy Criteria Based on the Jacobi Operator............ 129
6.5 Self-Adjoint Extensions of the Jacobi Operator............... 133
6.6 Markov’s Theorem............................................. 137
6.7 Continued Fractions ......................................... 140
6.8 Exercises.................................................... 143
6.9 Notes........................................................ 144
7 The Indeterminate Hamburger Moment Problem........................... 145
7.1 The Nevanlinna Functions A(z), B(z), C(z) D(z) ............. 145
7.2 Von Neumann Solutions...................................... 149
7.3 Weyl Circles................................................ 151
7.4 Nevanlinna Parametrization................................... 153
7.5 Maximal Point Masses....................................... 157
Contents ix
7.6 Nevanlinna-Pick Interpolation................................. 160
7.7 Solutions of Finite Order..................................... 165
7.8 Exercises................................................... 173
7.9 Notes......................................................... 174
8 The Operator-Theoretic Approach to the Stieltjes Moment
Problem.............................................................. 177
8.1 Preliminaries on Quadratic Forms on Hilbert Spaces............ 177
8.2 Existence of Solutions of the Stieltjes Moment Problem........ 179
8.3 Determinacy of the Stieltjes Moment Problem................... 180
8.4 Friedrichs and Krein Approximants............................. 184
8.5 Nevanlinna Parametrization for the Indeterminate Stieltjes
Moment Problem.............................................. 193
8.6 Weyl Circles for the Indeterminate Stieltjes Moment Problem... 196
8.7 Exercises..................................................... 199
8.8 Notes......................................................... 199
Part II The One-Dimensional Truncated Moment Problem
9 The One-Dimensional Truncated Hamburger and Stieltjes
Moment Problems...................................................... 203
9.1 Quadrature Formulas and the Truncated Moment Problem
for Positive Definite 2n-Sequences............................ 204
9.2 Hamburger’s Theorem and Markov’s Theorem Revisited........... 209
9.3 The Reproducing Kernel and the Christoffel Function........... 211
9.4 Positive Semidefinite 2n-*Sequences........................... 214
9.5 The Hankel Rank of a Positive Semidefinite 2rc-Sequence...... 217
9.6 Truncated Hamburger and Stieltjes Moment Sequences............ 221
9.7 Exercises................................................... 228
9.8 Notes......................................................... 228
10 The One-Dimensional Truncated Moment Problem on a
Bounded Interval................................................... 229
10.1 Existence of a Solution....................................... 229
10.2 The Moment Cone Sw+i and Its Boundary Points................. 231
10.3 Interior Points of Sw+i and Interlacing Properties of Roots. 234
10.4 Principal Measures of Interior Points of Sm+1 .............. 237
10.5 Maximal Masses and Canonical Measures...................... 241
10.6 A Parametrization of Canonical Measures....................... 246
10.7 Orthogonal Polynomials and Maximal Masses..................... 248
10.8 Exercises..................................................... 254
10.9 Notes......................................................... 255
11 The Moment Problem on the Unit Circle................... 257
11.1 The Fejér—Riesz Theorem....................................... 257
11.2 Trigonometric Moment Problem: Existence of a Solution........ 259
11.3 Orthogonal Polynomials on the Unit Circle..................... 261
x Contents
11.4 The Truncated Trigonometric Moment Problem................. 266
11.5 Caratheodory Functions, the Schur Algorithm,
and Geromimus* Theorem..................................... 271
11.6 Exercises.................................................. 278
11.7 Notes...................................................... 279
Part III The Multidimensional Moment Problem
12 The Moment Problem on Compact Semi-Algebraic Sets.................. 283
12.1 Semi-Algebraic Sets and Positivstellensatze................ 284
12.2 Localizing Functionals and Supports of Representing
Measures.................................................. 288
12.3 The Moment Problem on Compact Semi-Algebraic Sets
and the Strict Positivstellensatz.......................... 293
12.4 The Representation Theorem for Archimedean Modules......... 298
12.5 The Operator-Theoretic Approach to the Moment Problem______ 302
12.6 The Moment Problem for Semi-Algebraic Sets Contained
in Compact Polyhedra....................................... 307
12.7 Examples and Applications.................................. 308
12.8 Exercises.................................................. 311
12.9 Notes...................................................... 312
13 The Moment Problem on Closed Semi-Algebraic Sets: Existence_______ 315
13.1 Positive Polynomials and Sums of Squares................... 316
13.2 Properties (MP) and (SMP) ............................... 320
13.3 The Fibre Theorem.......................................... 322
13.4 (SMP) for Basic Closed Semi-Algebraic Subsets
of the Real Line........................................... 326
13.5 Application of the Fibre Theorem: Cylinder Sets
with Compact Base.......................................... 330
13.6 Application of the Fibre Theorem: The Rational Moment
Problem on R/*............................................. 332
13.7 Application of the Fibre Theorem: A Characterization
of Moment Functionals...................................... 335
13.8 Closedness and Stability of Quadratic Modules.............. 338
13.9 The Moment Problem on Some Cubics.......................... 344
13.10 Proofs of the Main Implications of Theorems 13.10
and 13.12.................................................. 348
13.11 Exercises.............................................. 353
13.12 Notes...................................................... 355
14 The Multidimensional Moment Problem: Determinacy................... 357
14.1 Various Notions of Determinacy........................... 357
14.2 Polynomial Approximation................................... 361
14.3 Partial Determinacy and Moment Functionals................. 364
14.4 The Multivariate Carleman Condition........................ 368
Contents xi
14.5 Moments of Gaussian Measure and Surface Measure
on the Unit Sphere......................................... 373
14.6 Disintegration Techniques and Determinacy.................. 374
14.7 Exercises................................................ 378
14.8 Notes...................................................... 380
15 The Complex Moment Problem........................................ 381
15.1 Relations Between Complex and Real Moment Problems........ 381
15.2 The Moment Problems for the ^-Semigroups
Z* and Wo x ............................................... 384
15.3 The Operator-Theoretic Approach to the Complex
Moment Problem............................................. 386
15.4 The Complex Carleman Condition............................. 391
15.5 An Extension Theorem for the Complex Moment Problem....... 393
15.6 The Two-Sided Complex Moment Problem....................... 395
15.7 Exercises.................................................. 397
15.8 Notes...................................................... 398
16 Semidefinite Programming and Polynomial Optimization.............. 399
16.1 Semidefinite Programming................................... 399
16.2 Lasserre Relaxations of Polynomial Optimization with
Constraints................................................ 403
16.3 Polynomial Optimization with Constraints................... 405
16.4 Global Optimization........................................ 408
16.5 Exercises.................................................. 410
16.6 Notes................................................... 411
Part IV The Multidimensional Truncated Moment Problem
17 Multidimensional Truncated Moment Problems: Existence............ 415
17.1 The Truncated /С-Moment Problem and Existence.............. 416
17.2 The Truncated Moment Problem on Projective Space........... 420
17.3 Hankel Matrices............................................ 425
17.4 Hankel Matrices of Functionals with Finitely Atomic
Measures................................................... 429
17.5 The Full Moment Problem with Finite Rank Hankel Matrix.... 433
17.6 Flat Extensions and the Flat Extension Theorem............. 434
17.7 Proof of Theorem 17.36 .................................... 438
17.8 Exercises................................................. 441
17.9 Notes...................................................... 442
18 Multidimensional Truncated Moment Problems: Basic
Concepts and Special Topics ...................................... 445
18.1 The Cone of Truncated Moment Functionals................... 445
18.2 Inner Moment Functionals and Boundary
Moment Functionals....................................... 451
18.3 The Core Variety of a Linear Functional.................... 453
xii
Contents
18.4 Maximal Masses..........................:................ 456
18.5 Constructing Ordered Maximal Mass Measures............... 462
18.6 Evaluation Polynomials................................. 464
18.7 Exercises................................................ 469
18.8 Notes.................................................... 470
19 The Truncated Moment Problem for Homogeneous Polynomials_________ 471
19.1 The Apolar Scalar Product.............................. 471
19.2 The Apolar Scalar Product and Differential Operators___... 475
19.3 The Apolar Scalar Product and the Truncated
Moment Problem .......................................... 477
19.4 Robinson’s Polynomial and Some Examples.................. 483
19.5 Zeros of Positive Homogeneous Polynomials................ 487
19.6 Applications to the Truncated Moment Problem on 1R2 ..... 491
19.7 Exercises................................................ 496
19.8 Notes.................................................... 497
Appendix........................................................... 499
A. 1 Measure Theory......................................... 499
A.2 Pick Functions and Stieltjes Transforms.................. 502
A.3 Positive Semidefinite and Positive Definite Matrices..... 504
A.4 Positive Semidefinite Block Matrices and Flat Extensions. 506
A.5 Locally Convex Topologies.............................. 509
A.6 Convex Sets and Cones.................................... 510
A.7 Symmetric and Self-Adjoint Operators on Hilbert Space.... 514
Bibliography......................................................... 517
Symbol Index......................................................... 527
Index
531
|
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| id | DE-604.BV044706073 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:59:54Z |
| institution | BVB |
| isbn | 9783319645452 3319645455 9783319878171 |
| language | English |
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| physical | xii, 535 Seiten Illustrationen |
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| series2 | Graduate texts in mathematics |
| spelling | Schmüdgen, Konrad 1947- Verfasser (DE-588)115774599 aut The moment problem Konrad Schmüdgen Cham Springer [2017] © 2017 xii, 535 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 277 Hier auch später erschienene, unveränderte Nachdrucke Literaturverzeichnis Seite 517-526 Momentenproblem (DE-588)4170422-8 gnd rswk-swf Momentenproblem (DE-588)4170422-8 s DE-604 Erscheint auch als Online-Ausgabe 978-3-319-64546-9 Graduate texts in mathematics 277 (DE-604)BV000000067 277 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030102696&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schmüdgen, Konrad 1947- The moment problem Graduate texts in mathematics Momentenproblem (DE-588)4170422-8 gnd |
| subject_GND | (DE-588)4170422-8 |
| title | The moment problem |
| title_auth | The moment problem |
| title_exact_search | The moment problem |
| title_full | The moment problem Konrad Schmüdgen |
| title_fullStr | The moment problem Konrad Schmüdgen |
| title_full_unstemmed | The moment problem Konrad Schmüdgen |
| title_short | The moment problem |
| title_sort | the moment problem |
| topic | Momentenproblem (DE-588)4170422-8 gnd |
| topic_facet | Momentenproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030102696&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT schmudgenkonrad themomentproblem |